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If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.. If (X, ≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions.
A ring is a set R equipped with two binary operations [a] + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms: [1] [2] [3] R is an abelian group under addition, meaning that: (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative). a + b = b + a for all a, b in R (that ...
For a semi-ring , the set of all finite unions of sets in is the ring generated by : = {: = =,} (One can show that () is equal to the set of all finite disjoint unions of sets in ). A content μ {\displaystyle \mu } defined on a semi-ring S {\displaystyle S} can be extended on the ring generated by S . {\displaystyle S.}
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the ๐-ring of .. By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a ๐-algebra.
δ-ring – Ring closed under countable intersections Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets Join (sigma algebra) – Algebraic structure of set algebra Pages displaying short descriptions of redirect targets
(In ring theory it is used for the exclusive or operation) ~ 1. The difference of two sets: x~y is the set of elements of x not in y. 2. An equivalence relation \ The difference of two sets: x\y is the set of elements of x not in y. − The difference of two sets: x−y is the set of elements of x not in y. ≈ Has the same cardinality as × A ...